solve: log8(w-6)=2-log8(w+15)
note: the 8 is to be little at the bottom right corner of log
since 64 = 8^2, then using base-8 logs, we have
log(w-6) = log(64)-log(w+15)
(w-6) = 64/(w+15)
(w-6)(w+15) = 64
w^2 + 9w - 154 = 0
w = 1/2 (-9±√697)
scrap the negative value (why?)
because the answer has got to be a real number which is not a negative number
how do you get 1/2 as a solution?
So technically there would be no solution right?
So, technically, the solution is
x = 1/2 (-9+√697) = 8.70
where does the 1/2 come from? Think back ... back ... back to Algebra I and the quadratic formula.
the quadratic formula is -b+-radical b^2-4ac/a
SO HOW DO YOU GET 1/2 FORM THAT?
Britt, when Steve and I went to school, the quadratic formula used to be
(-b ± √(b^2 -4ac)/(2a) or
(1/2) (-b ± √(b^2 -4ac)/a , thus the 1/2
I was not aware of any recent changes to the formula.
To solve the given equation:
log₈(w - 6) = 2 - log₈(w + 15)
Step 1: Combine the logarithms on the right side using the properties of logarithms.
log₈(w - 6) + log₈(w + 15) = 2
Step 2: Apply the logarithmic identity logₐ(b) + logₐ(c) = logₐ(b * c).
log₈((w - 6)(w + 15)) = 2
Step 3: Simplify the left side using the distributive property.
log₈(w² + 15w - 6w - 90) = 2
Step 4: Combine like terms inside the logarithm.
log₈(w² + 9w - 90) = 2
Step 5: Rewrite the logarithmic equation in exponential form.
w² + 9w - 90 = 8²
Step 6: Simplify the equation on the right.
w² + 9w - 90 = 64
Step 7: Move all terms to one side of the equation.
w² + 9w - 90 - 64 = 0
Step 8: Combine like terms.
w² + 9w - 154 = 0
Step 9: Factorize the quadratic equation.
(w - 7)(w + 22) = 0
Step 10: Set each factor equal to zero and solve for w.
w - 7 = 0, w + 22 = 0
w = 7, w = -22
So, the possible solutions to the equation are w = 7 and w = -22.